conica

Conica refers to conic sections in geometry: curves obtained by intersecting a plane with a cone. When a plane cuts a right circular cone, the intersection can be a circle, an ellipse, a parabola, or a hyperbola. Degenerate cases may produce a single point, a line, or two intersecting lines. The specific curve depends on the angle between the plane and the cone’s axis, or equivalently on the eccentricity of the resulting conic: a circle has eccentricity e = 0, an ellipse 0 < e < 1, a parabola e = 1, and a hyperbola e > 1.

Algebraically, a conic is any locus of points satisfying a second-degree equation in two variables: Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, with the discriminant B^2 − 4AC used to classify the type. Canonical forms describe conics in standard positions, such as circles and ellipses centered at the origin, or parabolas and hyperbolas aligned with the coordinate axes. Conics can also be described by a focus–directrix property or by a matrix form in projective geometry.

Key geometric properties include symmetry about axes, focal points (for ellipses and hyperbolas), and the directrix relation, as well as the role of the eccentricity in determining shape. Dandelin spheres provide a classical proof of the focus property for conic sections.

History and applications: conics were extensively studied by ancient Greek mathematicians, notably Apollonius of Perga, and later by Descartes. In modern science, conic sections model planetary orbits, optical reflections, architectural forms, and computer graphics, among other uses.

See also: conic sections. Note that Conica may also appear as a proper noun in other contexts, outside the geometric concept.